节: 使用平方差法进行分化 | 9.10 利用广场差异的因数化

章节大纲

Factorization Using Difference of Squares
::利用广场差异的因数化

When you learned how to multiply binomials we talked about two special products.
::当你学会如何乘以二元论时我们谈论了两种特殊产品

$\begin{aligned} The sum and difference formula: (a + b) (a - b) & = a^{2} - b^{2} \\ The square of a binomial formulas: (a + b)^{2} & = a^{2} + 2 a b + b^{2} \\ (a - b)^{2} & = a^{2} - 2 a b + b^{2} \end{aligned}$

::总和和差值公式(a+b)(a-b)=a2-b2 二元公式的平方a+b)2=a2+2ab+2ab+b2(a-b)2=a2-2ab+b2b2=a2-2ab+b2

In this section we’ll learn how to recognize and factor these special products.
::我们将学习如何认识这些特殊产品,

Factor the Difference of Two Squares
::两平方之差的因数

We use the sum and difference formula to factor a difference of two squares. A difference of two squares is any quadratic polynomial in the form $\begin{aligned} a^{2} - b^{2} \end{aligned}$ , where $\begin{aligned} a \end{aligned}$ and $\begin{aligned} b \end{aligned}$ can be variables, constants, or just about anything else. The factors of $\begin{aligned} a^{2} - b^{2} \end{aligned}$ are always $\begin{aligned} (a + b) (a - b) \end{aligned}$ ; the key is figuring out what the $\begin{aligned} a \end{aligned}$ and $\begin{aligned} b \end{aligned}$ terms are.
::我们用总和和差数公式来乘以两个正方形的差数。两个正方形的差数是A2-b2形式中的任何四边多边形,其中a和b可以是变量、常数,也可以是其他任何东西。a2-b2的因数总是(a+b)(a-b)-b;关键是确定a和b条件是什么。

Factoring the Difference of Squares
::计算平方之差

1. Factor the difference of squares :
::1. 将平方差乘以:

a) $\begin{aligned} x^{2} - 9 \end{aligned}$
::a) x2-9

Rewrite $\begin{aligned} x^{2} - 9 \end{aligned}$ as $\begin{aligned} x^{2} - 3^{2} \end{aligned}$ . Now it is obvious that it is a difference of squares.
::将 x2- 9 重写为 x2-32. 现在很明显,它是方形的差别。

The difference of squares formula is:
::方形公式的差别是:

$\begin{aligned} a^{2} - b^{2} = (a + b) (a - b) \end{aligned}$
::a2-b2=(a+b)(a-b)

Let’s see how our problem matches with the formula:
::让我们看看我们的问题如何与公式相吻合:

$\begin{aligned} x^{2} - 3^{2} = (x + 3) (x - 3) \end{aligned}$
::x2-32=(x+3)(x-3)

The answer is:
::答案是:

$\begin{aligned} x^{2} - 9 = (x + 3) (x - 3) \end{aligned}$
::x2- 9= (x+3)(x-3)

We can check to see if this is correct by multiplying $\begin{aligned} (x + 3) (x - 3) \end{aligned}$ :
::我们可以通过乘法( x+3)( x- 3) 检查是否正确 :

$\begin{aligned} x + 3 \\ \underline{x - 3} \\ - 3 x - 9 \\ \underline{x^{2} + 3 x} \\ x^{2} + 0 x - 9 \end{aligned}$

::x+3x-33x-9x2+3xxxx2+0x-9

The answer checks out.
::答案检查出来。

Note: We could factor this polynomial without recognizing it as a difference of squares. With the methods we learned in the last section we know that a quadratic polynomial factors into the product of two binomials:
::注意: 我们可以在不确认它为方形差异的情况下, 将这一多元性因素考虑在内。根据我们在最后一节中学到的方法, 我们知道,在两个二进制的产物中, 有一种四进制的多元性因素:

$\begin{aligned} (x) (x) \end{aligned}$

:x)(x)

We need to find two numbers that multiply to -9 and add to 0 (since there is no $\begin{aligned} x - \end{aligned}$ term , that’s the same as if the $\begin{aligned} x - \end{aligned}$ term had a coefficient of 0). We can write -9 as the following products:
::我们需要找到两个乘以 -9 并加到 0 的数字(因为没有x- term, 这与x- term 的系数值为 0 相同) 我们可以将 - 9 写成以下产品 :

$\begin{aligned} - 9 = - 1 \cdot 9 & and & - 1 + 9 = 8 \\ - 9 = 1 \cdot (- 9) & and & 1 + (- 9) = - 8 \\ - 9 = 3 \cdot (- 3) & and & 3 + (- 3) = 0 T h e s e a r e t h e c o r r e c t n u m b e r s . \end{aligned}$

::-=9=9=9=9=9=9=11=9=9=9=9=9=9=8=9=9=9=9=9=9=9=9=9=9=9=9=9=9=9=9=9=3=3=3=3=3=3=0=0为正确数字。

We can factor $\begin{aligned} x^{2} - 9 \end{aligned}$ as $\begin{aligned} (x + 3) (x - 3) \end{aligned}$ , which is the same answer as before. You can always factor using the methods you learned in the previous section, but recognizing special products helps you factor them faster.
::我们可以将x2- 9乘以( x+3)( x- 3) , 这和以前一样。您总是可以使用上一节中学习的方法来计算, 但承认特殊产品有助于您更快地计算。

b) $\begin{aligned} x^{2} - 100 \end{aligned}$
:b) x2-100

Rewrite $\begin{aligned} x^{2} - 100 \end{aligned}$ as $\begin{aligned} x^{2} - 10^{2} \end{aligned}$ . This factors as $\begin{aligned} (x + 10) (x - 10) \end{aligned}$ .
::重写 x2 - 100 的 x2 - 102. 这些系数为 (x+10) (x- 10) 。

c) $\begin{aligned} x^{2} - 1 \end{aligned}$
::c) x2-1

Rewrite $\begin{aligned} x^{2} - 1 \end{aligned}$ as $\begin{aligned} x^{2} - 1^{2} \end{aligned}$ . This factors as $\begin{aligned} (x + 1) (x - 1) \end{aligned}$ .
::将 x2 - 1 重写为 x2 - 12. 。该系数为 (x+1)(x- 1) 。

2. Factor the difference of squares:
::2. 将平方差乘以:

a) $\begin{aligned} 16 x^{2} - 25 \end{aligned}$
:a) 16x2-2-25

Rewrite $\begin{aligned} 16 x^{2} - 25 \end{aligned}$ as $\begin{aligned} (4 x)^{2} - 5^{2} \end{aligned}$ . This factors as $\begin{aligned} (4 x + 5) (4 x - 5) \end{aligned}$ .
::将 16x2-25 重写为 (4x)2-52。该系数为 (4x+5)(4x-5) 。

b) $\begin{aligned} 4 x^{2} - 81 \end{aligned}$
:b) 4x2-81

Rewrite $\begin{aligned} 4 x^{2} - 81 \end{aligned}$ as $\begin{aligned} (2 x)^{2} - 9^{2} \end{aligned}$ . This factors as $\begin{aligned} (2 x + 9) (2 x - 9) \end{aligned}$ .
::将 4x2-81 重写为 2x)2-92。该系数为 2x+9 (2x-9) 。

c) $\begin{aligned} 49 x^{2} - 64 \end{aligned}$
:c) 49x2-64

Rewrite $\begin{aligned} 49 x^{2} - 64 \end{aligned}$ as $\begin{aligned} (7 x)^{2} - 8^{2} \end{aligned}$ . This factors as $\begin{aligned} (7 x + 8) (7 x - 8) \end{aligned}$ .
::将49x2-64重写为(7x)2-82。这一系数为(7x+8)(7x-8)。

3. Factor the difference of squares:
::3. 将平方差乘以:

a) $\begin{aligned} x^{2} - y^{2} \end{aligned}$
::a) x2-y2

$\begin{aligned} x^{2} - y^{2} \end{aligned}$ factors as $\begin{aligned} (x + y) (x - y) \end{aligned}$ .
::x2-y2 系数为 (x+y)(x-y) 。

b) $\begin{aligned} 9 x^{2} - 4 y^{2} \end{aligned}$
:b) 9x2-4y2

Rewrite $\begin{aligned} 9 x^{2} - 4 y^{2} \end{aligned}$ as $\begin{aligned} (3 x)^{2} - (2 y)^{2} \end{aligned}$ . This factors as $\begin{aligned} (3 x + 2 y) (3 x - 2 y) \end{aligned}$ .
::将 9x2--4y2 重写为 (3x)2- (2y)2. 这个系数为 (3x+2y)(3x- 2y) 。

c) $\begin{aligned} x^{2} y^{2} - 1 \end{aligned}$
:c) x2y2-2-1

Rewrite $\begin{aligned} x^{2} y^{2} - 1 \end{aligned}$ as $\begin{aligned} (x y)^{2} - 1^{2} \end{aligned}$ . This factors as $\begin{aligned} (x y + 1) (x y - 1) \end{aligned}$ .
::将 x2y2- 1 重写为 (xy) 2-12。该系数为 (xy+1)(xy-1) 。

Examples
::实例

Factor the difference of squares:
::平方之差的乘数 :

Example 1
::例1

$\begin{aligned} x^{4} - 25 \end{aligned}$
::x4 - 25

Rewrite $\begin{aligned} x^{4} - 25 \end{aligned}$ as $\begin{aligned} (x^{2})^{2} - 5^{2} \end{aligned}$ . This factors as $\begin{aligned} (x^{2} + 5) (x^{2} - 5) \end{aligned}$ .
::将x4-25重写为(x2)2-2-52。此系数为(x2+5)(x2-5)。

Example 2
::例2

$\begin{aligned} 16 x^{4} - y^{2} \end{aligned}$
::16x4-y2 16x4-y2

Rewrite $\begin{aligned} 16 x^{4} - y^{2} \end{aligned}$ as $\begin{aligned} (4 x^{2})^{2} - y^{2} \end{aligned}$ . This factors as $\begin{aligned} (4 x^{2} + y) (4 x^{2} - y) \end{aligned}$ .
::将 16x4-y2 重写为 (4x2)2-2-y2. 这个系数为 (4x2+y)(4x2-y)。

Example 3
::例3

$\begin{aligned} x^{2} y^{8} - 64 z^{2} \end{aligned}$
::x2y8-64z2

Rewrite $\begin{aligned} x^{2} y^{4} - 64 z^{2} \end{aligned}$ as $\begin{aligned} (x y^{2})^{2} - (8 z)^{2} \end{aligned}$ . This factors as $\begin{aligned} (x y^{2} + 8 z) (x y^{2} - 8 z) \end{aligned}$ .
::将x2y4-64z2重写为(xy2)2-2-(8z)2. 该系数为(xy2+8z)(xy2-8z)。

Review
::回顾

Factor the following differences of squares.
::乘以下列方形差异。

$\begin{aligned} x^{2} - 4 \end{aligned}$
::x2 - 4

$\begin{aligned} x^{2} - 36 \end{aligned}$
::x2-36

$\begin{aligned} - x^{2} + 100 \end{aligned}$
::-x2+100

$\begin{aligned} x^{2} - 400 \end{aligned}$
::x2- 4000 x2 - 4000

$\begin{aligned} 9 x^{2} - 4 \end{aligned}$
::9x2-4

$\begin{aligned} 25 x^{2} - 49 \end{aligned}$
::25x2-49

$\begin{aligned} 9 a^{2} - 25 b^{2} \end{aligned}$
::9a2-25b2

$\begin{aligned} - 36 x^{2} + 25 \end{aligned}$
::- 36x2+25

$\begin{aligned} 4 x^{2} - y^{2} \end{aligned}$
::4x2 - Y2

$\begin{aligned} 16 x^{2} - 81 y^{2} \end{aligned}$
::16x2-81y2

Review (Answers)
::回顾(答复)

Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键。

章节大纲

Factorization Using Difference of Squares ::利用广场差异的因数化

Factor the Difference of Two Squares ::两平方之差的因数

Factoring the Difference of Squares ::计算平方之差

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Review ::回顾

Review (Answers) ::回顾(答复)

Factorization Using Difference of Squares
::利用广场差异的因数化

Factor the Difference of Two Squares
::两平方之差的因数

Factoring the Difference of Squares
::计算平方之差

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Review
::回顾

Review (Answers)
::回顾(答复)